To demonstrate the presence of quantum correlation, we reverse the polarizations of the control and probe in Ch0 as shown in Fig. REF  (b2). While the phase sensitive, non-reciprocal light transport persists under this new configuration (as we have confirmed experimentally [1]}), counter-intuitive quantum correlations now emerge.
To facilitate quantum measurements under the constraints of the relatively low optical depth and technical noise in the lasers, we switch off the input probe fields in all three channels, replacing them with coherent vacua.
We define the canonical position and momentum operators of the \(i\) th channel through Stokes operators: \(\hat{X}_i=\hat{S}_{x}^{i}/\sqrt{\left| S_{z}^{i} \right|}\)  and \(\hat{P}_i=\hat{S}_{y}^{i}/\sqrt{\left| S_{z}^{i} \right|}\) . The noise spectra \(\text{Var}(\hat{X}_{i})\)  and joint variance \(\text{Var}(\frac{\hat{X}_i\pm \hat{X}_j}{\sqrt{2}})\)  (\(i\) , \(j\) =0, 1, 2 and \(i\ne j\) ) of the steady state are then measured via joint polarization homodyne detections [2]}. For the detection, a bias magnetic field is applied along the propagation direction of light to shift the homodyne measurement from DC to the Larmor frequency (\(\sim \) 352 kHz), bypassing low-frequency technical noise. As illustrated in Fig. REF , quantum correlations manifest as \(\text{Var}(\frac{\hat{X}_i-\hat{X}_j}{\sqrt{2}})+\text{Var}(\frac{\hat{P}_i+\hat{P}_j}{\sqrt{2}})<\frac{\text{Var}(\hat{X}_{i})+\text{Var}(\hat{X}_{j})}{2}+\frac{\text{Var}(\hat{P}_{i})+\text{Var}(\hat{P}_{j})}{2}\) . To quantify the measured quantum correlation, we calculate the Gaussian discord \(\mathfrak {D}_{ij}\)  [3]}, [4]} at the Larmor frequency in the noise spectra between different channels. In the presence of a dissipative environment, Gaussian discord captures Gaussian quantum correlations, and is more robust than quantum entanglement in revealing quantum correlations. Following its definition, we have \(\mathfrak {D}_{01}=2.9\times 10^{-3}\) , \(\mathfrak {D}_{02}=2.5\times 10^{-3}\) , and \(\mathfrak {D}_{12}=2.5\times 10^{-3}\) . The positiveness of the evaluated discords unambiguously indicates the quantum nature of correlation between any two channels. The measured Gaussian discord is relatively low due to the small inter-channel coupling rate in our experiment, as significant information is lost to reservoir. For future studies, lasers with larger beam size and non-Gaussian profiles could be employed to enhance the quantum correlation between the optical channels.
